LMFDB - Embedded newform 4020.2.a.f.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1,4020)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 28x^{4} - 12x^{3} + 209x^{2} + 360x + 144 $ x^6 - x^5 - 28x^4 - 12x^3 + 209x^2 + 360x + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$4.67440$ of defining polynomial
Character	$\chi$	$=$	4020.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +4.67440 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} +4.67440 q^{7} +1.00000 q^{9} -0.759009 q^{11} -4.13882 q^{13} -1.00000 q^{15} +7.37981 q^{17} +4.28287 q^{19} -4.67440 q^{21} -1.29965 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.10971 q^{29} -0.705407 q^{31} +0.759009 q^{33} +4.67440 q^{35} +3.29965 q^{37} +4.13882 q^{39} +4.50124 q^{41} +0.367478 q^{43} +1.00000 q^{45} +0.932174 q^{47} +14.8500 q^{49} -7.37981 q^{51} -3.06593 q^{53} -0.759009 q^{55} -4.28287 q^{57} +6.13882 q^{59} -2.28287 q^{61} +4.67440 q^{63} -4.13882 q^{65} +1.00000 q^{67} +1.29965 q^{69} +6.64006 q^{71} -6.39258 q^{73} -1.00000 q^{75} -3.54792 q^{77} +5.62223 q^{79} +1.00000 q^{81} -12.1829 q^{83} +7.37981 q^{85} +2.10971 q^{87} -11.9992 q^{89} -19.3465 q^{91} +0.705407 q^{93} +4.28287 q^{95} +19.2317 q^{97} -0.759009 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{5} + q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 6 * q^5 + q^7 + 6 * q^9	$ 6 q - 6 q^{3} + 6 q^{5} + q^{7} + 6 q^{9} - 7 q^{11} + 7 q^{13} - 6 q^{15} + 10 q^{17} - 3 q^{19} - q^{21} - 4 q^{23} + 6 q^{25} - 6 q^{27} + 9 q^{29} + 3 q^{31} + 7 q^{33} + q^{35} + 16 q^{37} - 7 q^{39} + 7 q^{41} + 3 q^{43} + 6 q^{45} + q^{47} + 15 q^{49} - 10 q^{51} + 7 q^{53} - 7 q^{55} + 3 q^{57} + 5 q^{59} + 15 q^{61} + q^{63} + 7 q^{65} + 6 q^{67} + 4 q^{69} - 12 q^{71} + 12 q^{73} - 6 q^{75} + 9 q^{77} + 9 q^{79} + 6 q^{81} - 2 q^{83} + 10 q^{85} - 9 q^{87} + 10 q^{89} - 5 q^{91} - 3 q^{93} - 3 q^{95} + 37 q^{97} - 7 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^5 + q^7 + 6 * q^9 - 7 * q^11 + 7 * q^13 - 6 * q^15 + 10 * q^17 - 3 * q^19 - q^21 - 4 * q^23 + 6 * q^25 - 6 * q^27 + 9 * q^29 + 3 * q^31 + 7 * q^33 + q^35 + 16 * q^37 - 7 * q^39 + 7 * q^41 + 3 * q^43 + 6 * q^45 + q^47 + 15 * q^49 - 10 * q^51 + 7 * q^53 - 7 * q^55 + 3 * q^57 + 5 * q^59 + 15 * q^61 + q^63 + 7 * q^65 + 6 * q^67 + 4 * q^69 - 12 * q^71 + 12 * q^73 - 6 * q^75 + 9 * q^77 + 9 * q^79 + 6 * q^81 - 2 * q^83 + 10 * q^85 - 9 * q^87 + 10 * q^89 - 5 * q^91 - 3 * q^93 - 3 * q^95 + 37 * q^97 - 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.67440	1.76676	0.883379	−	0.468659i	$-0.155263\pi$
$7$	4.67440	1.76676	0.883379	+	0.468659i	$0.155263\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.759009	−0.228850	−0.114425	−	0.993432i	$-0.536503\pi$
$11$	−0.759009	−0.228850	−0.114425	+	0.993432i	$0.536503\pi$
$12$	0	0
$13$	−4.13882	−1.14790	−0.573951	−	0.818890i	$-0.694590\pi$
$13$	−4.13882	−1.14790	−0.573951	+	0.818890i	$0.694590\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	7.37981	1.78987	0.894933	−	0.446200i	$-0.147223\pi$
$17$	7.37981	1.78987	0.894933	+	0.446200i	$0.147223\pi$
$18$	0	0
$19$	4.28287	0.982558	0.491279	−	0.871002i	$-0.336530\pi$
$19$	4.28287	0.982558	0.491279	+	0.871002i	$0.336530\pi$
$20$	0	0
$21$	−4.67440	−1.02004
$22$	0	0
$23$	−1.29965	−0.270996	−0.135498	−	0.990778i	$-0.543263\pi$
$23$	−1.29965	−0.270996	−0.135498	+	0.990778i	$0.543263\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.10971	−0.391763	−0.195881	−	0.980628i	$-0.562757\pi$
$29$	−2.10971	−0.391763	−0.195881	+	0.980628i	$0.562757\pi$
$30$	0	0
$31$	−0.705407	−0.126695	−0.0633474	−	0.997992i	$-0.520178\pi$
$31$	−0.705407	−0.126695	−0.0633474	+	0.997992i	$0.520178\pi$
$32$	0	0
$33$	0.759009	0.132127
$34$	0	0
$35$	4.67440	0.790118
$36$	0	0
$37$	3.29965	0.542459	0.271230	−	0.962515i	$-0.412570\pi$
$37$	3.29965	0.542459	0.271230	+	0.962515i	$0.412570\pi$
$38$	0	0
$39$	4.13882	0.662742
$40$	0	0
$41$	4.50124	0.702975	0.351488	−	0.936193i	$-0.385676\pi$
$41$	4.50124	0.702975	0.351488	+	0.936193i	$0.385676\pi$
$42$	0	0
$43$	0.367478	0.0560398	0.0280199	−	0.999607i	$-0.491080\pi$
$43$	0.367478	0.0560398	0.0280199	+	0.999607i	$0.491080\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	0.932174	0.135972	0.0679858	−	0.997686i	$-0.478343\pi$
$47$	0.932174	0.135972	0.0679858	+	0.997686i	$0.478343\pi$
$48$	0	0
$49$	14.8500	2.12143
$50$	0	0
$51$	−7.37981	−1.03338
$52$	0	0
$53$	−3.06593	−0.421139	−0.210569	−	0.977579i	$-0.567532\pi$
$53$	−3.06593	−0.421139	−0.210569	+	0.977579i	$0.567532\pi$
$54$	0	0
$55$	−0.759009	−0.102345
$56$	0	0
$57$	−4.28287	−0.567280
$58$	0	0
$59$	6.13882	0.799206	0.399603	−	0.916688i	$-0.369148\pi$
$59$	6.13882	0.799206	0.399603	+	0.916688i	$0.369148\pi$
$60$	0	0
$61$	−2.28287	−0.292292	−0.146146	−	0.989263i	$-0.546687\pi$
$61$	−2.28287	−0.292292	−0.146146	+	0.989263i	$0.546687\pi$
$62$	0	0
$63$	4.67440	0.588919
$64$	0	0
$65$	−4.13882	−0.513357
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0	0
$69$	1.29965	0.156460
$70$	0	0
$71$	6.64006	0.788030	0.394015	−	0.919104i	$-0.371086\pi$
$71$	6.64006	0.788030	0.394015	+	0.919104i	$0.371086\pi$
$72$	0	0
$73$	−6.39258	−0.748195	−0.374097	−	0.927389i	$-0.622047\pi$
$73$	−6.39258	−0.748195	−0.374097	+	0.927389i	$0.622047\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−3.54792	−0.404323
$78$	0	0
$79$	5.62223	0.632550	0.316275	−	0.948667i	$-0.397568\pi$
$79$	5.62223	0.632550	0.316275	+	0.948667i	$0.397568\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.1829	−1.33725	−0.668624	−	0.743600i	$-0.733117\pi$
$83$	−12.1829	−1.33725	−0.668624	+	0.743600i	$0.733117\pi$
$84$	0	0
$85$	7.37981	0.800453
$86$	0	0
$87$	2.10971	0.226184
$88$	0	0
$89$	−11.9992	−1.27191	−0.635954	−	0.771727i	$-0.719393\pi$
$89$	−11.9992	−1.27191	−0.635954	+	0.771727i	$0.719393\pi$
$90$	0	0
$91$	−19.3465	−2.02807
$92$	0	0
$93$	0.705407	0.0731473
$94$	0	0
$95$	4.28287	0.439413
$96$	0	0
$97$	19.2317	1.95269	0.976344	−	0.216223i	$-0.0693739\pi$
$97$	19.2317	1.95269	0.976344	+	0.216223i	$0.0693739\pi$
$98$	0	0
$99$	−0.759009	−0.0762833
$100$	0	0
$101$	−2.08272	−0.207238	−0.103619	−	0.994617i	$-0.533042\pi$
$101$	−2.08272	−0.207238	−0.103619	+	0.994617i	$0.533042\pi$
$102$	0	0
$103$	−0.231827	−0.0228426	−0.0114213	−	0.999935i	$-0.503636\pi$
$103$	−0.231827	−0.0228426	−0.0114213	+	0.999935i	$0.503636\pi$
$104$	0	0
$105$	−4.67440	−0.456175
$106$	0	0
$107$	−3.74223	−0.361775	−0.180887	−	0.983504i	$-0.557897\pi$
$107$	−3.74223	−0.361775	−0.180887	+	0.983504i	$0.557897\pi$
$108$	0	0
$109$	15.2161	1.45744	0.728719	−	0.684813i	$-0.240116\pi$
$109$	15.2161	1.45744	0.728719	+	0.684813i	$0.240116\pi$
$110$	0	0
$111$	−3.29965	−0.313189
$112$	0	0
$113$	−11.6870	−1.09942	−0.549710	−	0.835355i	$-0.685262\pi$
$113$	−11.6870	−1.09942	−0.549710	+	0.835355i	$0.685262\pi$
$114$	0	0
$115$	−1.29965	−0.121193
$116$	0	0
$117$	−4.13882	−0.382634
$118$	0	0
$119$	34.4962	3.16226
$120$	0	0
$121$	−10.4239	−0.947628
$122$	0	0
$123$	−4.50124	−0.405863
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	10.4527	0.927527	0.463763	−	0.885959i	$-0.346499\pi$
$127$	10.4527	0.927527	0.463763	+	0.885959i	$0.346499\pi$
$128$	0	0
$129$	−0.367478	−0.0323546
$130$	0	0
$131$	−9.37981	−0.819518	−0.409759	−	0.912194i	$-0.634387\pi$
$131$	−9.37981	−0.819518	−0.409759	+	0.912194i	$0.634387\pi$
$132$	0	0
$133$	20.0199	1.73594
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−9.30026	−0.794575	−0.397288	−	0.917694i	$-0.630048\pi$
$137$	−9.30026	−0.794575	−0.397288	+	0.917694i	$0.630048\pi$
$138$	0	0
$139$	−19.9648	−1.69339	−0.846697	−	0.532076i	$-0.821412\pi$
$139$	−19.9648	−1.69339	−0.846697	+	0.532076i	$0.821412\pi$
$140$	0	0
$141$	−0.932174	−0.0785032
$142$	0	0
$143$	3.14140	0.262697
$144$	0	0
$145$	−2.10971	−0.175202
$146$	0	0
$147$	−14.8500	−1.22481
$148$	0	0
$149$	6.27275	0.513884	0.256942	−	0.966427i	$-0.417285\pi$
$149$	6.27275	0.513884	0.256942	+	0.966427i	$0.417285\pi$
$150$	0	0
$151$	6.68496	0.544014	0.272007	−	0.962295i	$-0.412313\pi$
$151$	6.68496	0.544014	0.272007	+	0.962295i	$0.412313\pi$
$152$	0	0
$153$	7.37981	0.596622
$154$	0	0
$155$	−0.705407	−0.0566597
$156$	0	0
$157$	9.82160	0.783849	0.391925	−	0.919997i	$-0.371809\pi$
$157$	9.82160	0.783849	0.391925	+	0.919997i	$0.371809\pi$
$158$	0	0
$159$	3.06593	0.243144
$160$	0	0
$161$	−6.07510	−0.478785
$162$	0	0
$163$	11.0419	0.864867	0.432433	−	0.901666i	$-0.357655\pi$
$163$	11.0419	0.864867	0.432433	+	0.901666i	$0.357655\pi$
$164$	0	0
$165$	0.759009	0.0590888
$166$	0	0
$167$	6.38964	0.494445	0.247223	−	0.968959i	$-0.420482\pi$
$167$	6.38964	0.494445	0.247223	+	0.968959i	$0.420482\pi$
$168$	0	0
$169$	4.12983	0.317679
$170$	0	0
$171$	4.28287	0.327519
$172$	0	0
$173$	−6.13693	−0.466582	−0.233291	−	0.972407i	$-0.574949\pi$
$173$	−6.13693	−0.466582	−0.233291	+	0.972407i	$0.574949\pi$
$174$	0	0
$175$	4.67440	0.353352
$176$	0	0
$177$	−6.13882	−0.461422
$178$	0	0
$179$	−5.95000	−0.444724	−0.222362	−	0.974964i	$-0.571377\pi$
$179$	−5.95000	−0.444724	−0.222362	+	0.974964i	$0.571377\pi$
$180$	0	0
$181$	15.2055	1.13022	0.565109	−	0.825016i	$-0.308834\pi$
$181$	15.2055	1.13022	0.565109	+	0.825016i	$0.308834\pi$
$182$	0	0
$183$	2.28287	0.168755
$184$	0	0
$185$	3.29965	0.242595
$186$	0	0
$187$	−5.60135	−0.409611
$188$	0	0
$189$	−4.67440	−0.340013
$190$	0	0
$191$	−17.3951	−1.25866	−0.629331	−	0.777137i	$-0.716671\pi$
$191$	−17.3951	−1.25866	−0.629331	+	0.777137i	$0.716671\pi$
$192$	0	0
$193$	18.8082	1.35384	0.676921	−	0.736056i	$-0.263314\pi$
$193$	18.8082	1.35384	0.676921	+	0.736056i	$0.263314\pi$
$194$	0	0
$195$	4.13882	0.296387
$196$	0	0
$197$	9.50851	0.677453	0.338727	−	0.940885i	$-0.390004\pi$
$197$	9.50851	0.677453	0.338727	+	0.940885i	$0.390004\pi$
$198$	0	0
$199$	−21.1613	−1.50009	−0.750044	−	0.661388i	$-0.769968\pi$
$199$	−21.1613	−1.50009	−0.750044	+	0.661388i	$0.769968\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	0	0
$203$	−9.86162	−0.692150
$204$	0	0
$205$	4.50124	0.314380
$206$	0	0
$207$	−1.29965	−0.0903321
$208$	0	0
$209$	−3.25074	−0.224858
$210$	0	0
$211$	−5.73164	−0.394582	−0.197291	−	0.980345i	$-0.563214\pi$
$211$	−5.73164	−0.394582	−0.197291	+	0.980345i	$0.563214\pi$
$212$	0	0
$213$	−6.64006	−0.454969
$214$	0	0
$215$	0.367478	0.0250618
$216$	0	0
$217$	−3.29736	−0.223839
$218$	0	0
$219$	6.39258	0.431970
$220$	0	0
$221$	−30.5437	−2.05459
$222$	0	0
$223$	12.6470	0.846906	0.423453	−	0.905918i	$-0.360818\pi$
$223$	12.6470	0.846906	0.423453	+	0.905918i	$0.360818\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	10.3426	0.686464	0.343232	−	0.939251i	$-0.388478\pi$
$227$	10.3426	0.686464	0.343232	+	0.939251i	$0.388478\pi$
$228$	0	0
$229$	−6.47195	−0.427679	−0.213839	−	0.976869i	$-0.568597\pi$
$229$	−6.47195	−0.427679	−0.213839	+	0.976869i	$0.568597\pi$
$230$	0	0
$231$	3.54792	0.233436
$232$	0	0
$233$	25.3946	1.66366	0.831828	−	0.555033i	$-0.187294\pi$
$233$	25.3946	1.66366	0.831828	+	0.555033i	$0.187294\pi$
$234$	0	0
$235$	0.932174	0.0608084
$236$	0	0
$237$	−5.62223	−0.365203
$238$	0	0
$239$	−21.7439	−1.40650	−0.703248	−	0.710945i	$-0.748267\pi$
$239$	−21.7439	−1.40650	−0.703248	+	0.710945i	$0.748267\pi$
$240$	0	0
$241$	28.2735	1.82125	0.910627	−	0.413230i	$-0.135599\pi$
$241$	28.2735	1.82125	0.910627	+	0.413230i	$0.135599\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	14.8500	0.948735
$246$	0	0
$247$	−17.7260	−1.12788
$248$	0	0
$249$	12.1829	0.772061
$250$	0	0
$251$	15.6216	0.986030	0.493015	−	0.870021i	$-0.335895\pi$
$251$	15.6216	0.986030	0.493015	+	0.870021i	$0.335895\pi$
$252$	0	0
$253$	0.986448	0.0620175
$254$	0	0
$255$	−7.37981	−0.462142
$256$	0	0
$257$	18.5901	1.15962	0.579808	−	0.814753i	$-0.303128\pi$
$257$	18.5901	1.15962	0.579808	+	0.814753i	$0.303128\pi$
$258$	0	0
$259$	15.4239	0.958395
$260$	0	0
$261$	−2.10971	−0.130588
$262$	0	0
$263$	−2.37571	−0.146492	−0.0732461	−	0.997314i	$-0.523336\pi$
$263$	−2.37571	−0.146492	−0.0732461	+	0.997314i	$0.523336\pi$
$264$	0	0
$265$	−3.06593	−0.188339
$266$	0	0
$267$	11.9992	0.734336
$268$	0	0
$269$	−1.47818	−0.0901261	−0.0450631	−	0.998984i	$-0.514349\pi$
$269$	−1.47818	−0.0901261	−0.0450631	+	0.998984i	$0.514349\pi$
$270$	0	0
$271$	2.44807	0.148710	0.0743549	−	0.997232i	$-0.476310\pi$
$271$	2.44807	0.148710	0.0743549	+	0.997232i	$0.476310\pi$
$272$	0	0
$273$	19.3465	1.17090
$274$	0	0
$275$	−0.759009	−0.0457700
$276$	0	0
$277$	3.66859	0.220424	0.110212	−	0.993908i	$-0.464847\pi$
$277$	3.66859	0.220424	0.110212	+	0.993908i	$0.464847\pi$
$278$	0	0
$279$	−0.705407	−0.0422316
$280$	0	0
$281$	22.6308	1.35004	0.675021	−	0.737799i	$-0.264135\pi$
$281$	22.6308	1.35004	0.675021	+	0.737799i	$0.264135\pi$
$282$	0	0
$283$	2.84696	0.169234	0.0846170	−	0.996414i	$-0.473033\pi$
$283$	2.84696	0.169234	0.0846170	+	0.996414i	$0.473033\pi$
$284$	0	0
$285$	−4.28287	−0.253695
$286$	0	0
$287$	21.0406	1.24199
$288$	0	0
$289$	37.4616	2.20362
$290$	0	0
$291$	−19.2317	−1.12738
$292$	0	0
$293$	2.75351	0.160862	0.0804310	−	0.996760i	$-0.474370\pi$
$293$	2.75351	0.160862	0.0804310	+	0.996760i	$0.474370\pi$
$294$	0	0
$295$	6.13882	0.357416
$296$	0	0
$297$	0.759009	0.0440422
$298$	0	0
$299$	5.37903	0.311077
$300$	0	0
$301$	1.71774	0.0990089
$302$	0	0
$303$	2.08272	0.119649
$304$	0	0
$305$	−2.28287	−0.130717
$306$	0	0
$307$	2.39592	0.136743	0.0683713	−	0.997660i	$-0.478220\pi$
$307$	2.39592	0.136743	0.0683713	+	0.997660i	$0.478220\pi$
$308$	0	0
$309$	0.231827	0.0131882
$310$	0	0
$311$	24.6362	1.39699	0.698496	−	0.715614i	$-0.253853\pi$
$311$	24.6362	1.39699	0.698496	+	0.715614i	$0.253853\pi$
$312$	0	0
$313$	12.3413	0.697570	0.348785	−	0.937203i	$-0.386594\pi$
$313$	12.3413	0.697570	0.348785	+	0.937203i	$0.386594\pi$
$314$	0	0
$315$	4.67440	0.263373
$316$	0	0
$317$	12.7136	0.714069	0.357034	−	0.934091i	$-0.383788\pi$
$317$	12.7136	0.714069	0.357034	+	0.934091i	$0.383788\pi$
$318$	0	0
$319$	1.60129	0.0896549
$320$	0	0
$321$	3.74223	0.208871
$322$	0	0
$323$	31.6068	1.75865
$324$	0	0
$325$	−4.13882	−0.229580
$326$	0	0
$327$	−15.2161	−0.841452
$328$	0	0
$329$	4.35736	0.240229
$330$	0	0
$331$	−16.4136	−0.902175	−0.451088	−	0.892480i	$-0.648964\pi$
$331$	−16.4136	−0.902175	−0.451088	+	0.892480i	$0.648964\pi$
$332$	0	0
$333$	3.29965	0.180820
$334$	0	0
$335$	1.00000	0.0546358
$336$	0	0
$337$	20.5048	1.11697	0.558485	−	0.829515i	$-0.311383\pi$
$337$	20.5048	1.11697	0.558485	+	0.829515i	$0.311383\pi$
$338$	0	0
$339$	11.6870	0.634751
$340$	0	0
$341$	0.535411	0.0289941
$342$	0	0
$343$	36.6943	1.98130
$344$	0	0
$345$	1.29965	0.0699709
$346$	0	0
$347$	−28.3369	−1.52121	−0.760603	−	0.649218i	$-0.775096\pi$
$347$	−28.3369	−1.52121	−0.760603	+	0.649218i	$0.775096\pi$
$348$	0	0
$349$	13.7899	0.738154	0.369077	−	0.929399i	$-0.379674\pi$
$349$	13.7899	0.738154	0.369077	+	0.929399i	$0.379674\pi$
$350$	0	0
$351$	4.13882	0.220914
$352$	0	0
$353$	−14.5274	−0.773215	−0.386607	−	0.922244i	$-0.626353\pi$
$353$	−14.5274	−0.773215	−0.386607	+	0.922244i	$0.626353\pi$
$354$	0	0
$355$	6.64006	0.352418
$356$	0	0
$357$	−34.4962	−1.82573
$358$	0	0
$359$	31.0361	1.63802	0.819010	−	0.573779i	$-0.194523\pi$
$359$	31.0361	1.63802	0.819010	+	0.573779i	$0.194523\pi$
$360$	0	0
$361$	−0.657011	−0.0345795
$362$	0	0
$363$	10.4239	0.547113
$364$	0	0
$365$	−6.39258	−0.334603
$366$	0	0
$367$	−14.2186	−0.742203	−0.371102	−	0.928592i	$-0.621020\pi$
$367$	−14.2186	−0.742203	−0.371102	+	0.928592i	$0.621020\pi$
$368$	0	0
$369$	4.50124	0.234325
$370$	0	0
$371$	−14.3314	−0.744050
$372$	0	0
$373$	30.2211	1.56479	0.782395	−	0.622782i	$-0.213998\pi$
$373$	30.2211	1.56479	0.782395	+	0.622782i	$0.213998\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	8.73170	0.449705
$378$	0	0
$379$	10.9686	0.563420	0.281710	−	0.959500i	$-0.409098\pi$
$379$	10.9686	0.563420	0.281710	+	0.959500i	$0.409098\pi$
$380$	0	0
$381$	−10.4527	−0.535508
$382$	0	0
$383$	−1.51331	−0.0773264	−0.0386632	−	0.999252i	$-0.512310\pi$
$383$	−1.51331	−0.0773264	−0.0386632	+	0.999252i	$0.512310\pi$
$384$	0	0
$385$	−3.54792	−0.180819
$386$	0	0
$387$	0.367478	0.0186799
$388$	0	0
$389$	−4.89088	−0.247977	−0.123989	−	0.992284i	$-0.539569\pi$
$389$	−4.89088	−0.247977	−0.123989	+	0.992284i	$0.539569\pi$
$390$	0	0
$391$	−9.59119	−0.485047
$392$	0	0
$393$	9.37981	0.473149
$394$	0	0
$395$	5.62223	0.282885
$396$	0	0
$397$	−20.0040	−1.00397	−0.501987	−	0.864875i	$-0.667397\pi$
$397$	−20.0040	−1.00397	−0.501987	+	0.864875i	$0.667397\pi$
$398$	0	0
$399$	−20.0199	−1.00225
$400$	0	0
$401$	−7.91394	−0.395203	−0.197602	−	0.980282i	$-0.563315\pi$
$401$	−7.91394	−0.395203	−0.197602	+	0.980282i	$0.563315\pi$
$402$	0	0
$403$	2.91955	0.145433
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−2.50447	−0.124142
$408$	0	0
$409$	−30.9027	−1.52804	−0.764019	−	0.645194i	$-0.776777\pi$
$409$	−30.9027	−1.52804	−0.764019	+	0.645194i	$0.776777\pi$
$410$	0	0
$411$	9.30026	0.458748
$412$	0	0
$413$	28.6953	1.41200
$414$	0	0
$415$	−12.1829	−0.598036
$416$	0	0
$417$	19.9648	0.977681
$418$	0	0
$419$	−37.5739	−1.83560	−0.917802	−	0.397038i	$-0.870038\pi$
$419$	−37.5739	−1.83560	−0.917802	+	0.397038i	$0.870038\pi$
$420$	0	0
$421$	10.6817	0.520594	0.260297	−	0.965529i	$-0.416179\pi$
$421$	10.6817	0.520594	0.260297	+	0.965529i	$0.416179\pi$
$422$	0	0
$423$	0.932174	0.0453239
$424$	0	0
$425$	7.37981	0.357973
$426$	0	0
$427$	−10.6711	−0.516409
$428$	0	0
$429$	−3.14140	−0.151668
$430$	0	0
$431$	23.6479	1.13908	0.569541	−	0.821963i	$-0.307121\pi$
$431$	23.6479	1.13908	0.569541	+	0.821963i	$0.307121\pi$
$432$	0	0
$433$	−2.94166	−0.141367	−0.0706836	−	0.997499i	$-0.522518\pi$
$433$	−2.94166	−0.141367	−0.0706836	+	0.997499i	$0.522518\pi$
$434$	0	0
$435$	2.10971	0.101153
$436$	0	0
$437$	−5.56624	−0.266270
$438$	0	0
$439$	−17.0228	−0.812455	−0.406228	−	0.913772i	$-0.633156\pi$
$439$	−17.0228	−0.812455	−0.406228	+	0.913772i	$0.633156\pi$
$440$	0	0
$441$	14.8500	0.707145
$442$	0	0
$443$	−16.6404	−0.790610	−0.395305	−	0.918550i	$-0.629361\pi$
$443$	−16.6404	−0.790610	−0.395305	+	0.918550i	$0.629361\pi$
$444$	0	0
$445$	−11.9992	−0.568815
$446$	0	0
$447$	−6.27275	−0.296691
$448$	0	0
$449$	7.71774	0.364223	0.182111	−	0.983278i	$-0.441707\pi$
$449$	7.71774	0.364223	0.182111	+	0.983278i	$0.441707\pi$
$450$	0	0
$451$	−3.41648	−0.160876
$452$	0	0
$453$	−6.68496	−0.314087
$454$	0	0
$455$	−19.3465	−0.906978
$456$	0	0
$457$	6.76424	0.316418	0.158209	−	0.987406i	$-0.449428\pi$
$457$	6.76424	0.316418	0.158209	+	0.987406i	$0.449428\pi$
$458$	0	0
$459$	−7.37981	−0.344460
$460$	0	0
$461$	−16.4541	−0.766346	−0.383173	−	0.923677i	$-0.625169\pi$
$461$	−16.4541	−0.766346	−0.383173	+	0.923677i	$0.625169\pi$
$462$	0	0
$463$	17.1279	0.796003	0.398002	−	0.917385i	$-0.369704\pi$
$463$	17.1279	0.796003	0.398002	+	0.917385i	$0.369704\pi$
$464$	0	0
$465$	0.705407	0.0327125
$466$	0	0
$467$	−18.9265	−0.875814	−0.437907	−	0.899020i	$-0.644280\pi$
$467$	−18.9265	−0.875814	−0.437907	+	0.899020i	$0.644280\pi$
$468$	0	0
$469$	4.67440	0.215844
$470$	0	0
$471$	−9.82160	−0.452556
$472$	0	0
$473$	−0.278919	−0.0128247
$474$	0	0
$475$	4.28287	0.196512
$476$	0	0
$477$	−3.06593	−0.140380
$478$	0	0
$479$	−18.3082	−0.836523	−0.418262	−	0.908327i	$-0.637360\pi$
$479$	−18.3082	−0.836523	−0.418262	+	0.908327i	$0.637360\pi$
$480$	0	0
$481$	−13.6567	−0.622690
$482$	0	0
$483$	6.07510	0.276427
$484$	0	0
$485$	19.2317	0.873269
$486$	0	0
$487$	−26.0382	−1.17990	−0.589952	−	0.807439i	$-0.700853\pi$
$487$	−26.0382	−1.17990	−0.589952	+	0.807439i	$0.700853\pi$
$488$	0	0
$489$	−11.0419	−0.499331
$490$	0	0
$491$	−30.8613	−1.39275	−0.696376	−	0.717677i	$-0.745205\pi$
$491$	−30.8613	−1.39275	−0.696376	+	0.717677i	$0.745205\pi$
$492$	0	0
$493$	−15.5692	−0.701203
$494$	0	0
$495$	−0.759009	−0.0341149
$496$	0	0
$497$	31.0383	1.39226
$498$	0	0
$499$	−39.8833	−1.78542	−0.892710	−	0.450631i	$-0.851199\pi$
$499$	−39.8833	−1.78542	−0.892710	+	0.450631i	$0.851199\pi$
$500$	0	0
$501$	−6.38964	−0.285468
$502$	0	0
$503$	−26.0574	−1.16184	−0.580921	−	0.813960i	$-0.697308\pi$
$503$	−26.0574	−1.16184	−0.580921	+	0.813960i	$0.697308\pi$
$504$	0	0
$505$	−2.08272	−0.0926796
$506$	0	0
$507$	−4.12983	−0.183412
$508$	0	0
$509$	12.0411	0.533712	0.266856	−	0.963736i	$-0.414015\pi$
$509$	12.0411	0.533712	0.266856	+	0.963736i	$0.414015\pi$
$510$	0	0
$511$	−29.8815	−1.32188
$512$	0	0
$513$	−4.28287	−0.189093
$514$	0	0
$515$	−0.231827	−0.0102155
$516$	0	0
$517$	−0.707529	−0.0311171
$518$	0	0
$519$	6.13693	0.269381
$520$	0	0
$521$	−12.7773	−0.559784	−0.279892	−	0.960032i	$-0.590299\pi$
$521$	−12.7773	−0.559784	−0.279892	+	0.960032i	$0.590299\pi$
$522$	0	0
$523$	−38.4983	−1.68341	−0.841706	−	0.539936i	$-0.818448\pi$
$523$	−38.4983	−1.68341	−0.841706	+	0.539936i	$0.818448\pi$
$524$	0	0
$525$	−4.67440	−0.204008
$526$	0	0
$527$	−5.20577	−0.226767
$528$	0	0
$529$	−21.3109	−0.926561
$530$	0	0
$531$	6.13882	0.266402
$532$	0	0
$533$	−18.6298	−0.806947
$534$	0	0
$535$	−3.74223	−0.161791
$536$	0	0
$537$	5.95000	0.256762
$538$	0	0
$539$	−11.2713	−0.485490
$540$	0	0
$541$	6.30365	0.271015	0.135508	−	0.990776i	$-0.456734\pi$
$541$	6.30365	0.271015	0.135508	+	0.990776i	$0.456734\pi$
$542$	0	0
$543$	−15.2055	−0.652532
$544$	0	0
$545$	15.2161	0.651786
$546$	0	0
$547$	0.103389	0.00442058	0.00221029	−	0.999998i	$-0.499296\pi$
$547$	0.103389	0.00442058	0.00221029	+	0.999998i	$0.499296\pi$
$548$	0	0
$549$	−2.28287	−0.0974306
$550$	0	0
$551$	−9.03560	−0.384930
$552$	0	0
$553$	26.2806	1.11756
$554$	0	0
$555$	−3.29965	−0.140062
$556$	0	0
$557$	43.0950	1.82599	0.912997	−	0.407966i	$-0.133762\pi$
$557$	43.0950	1.82599	0.912997	+	0.407966i	$0.133762\pi$
$558$	0	0
$559$	−1.52092	−0.0643283
$560$	0	0
$561$	5.60135	0.236489
$562$	0	0
$563$	4.51866	0.190439	0.0952193	−	0.995456i	$-0.469645\pi$
$563$	4.51866	0.190439	0.0952193	+	0.995456i	$0.469645\pi$
$564$	0	0
$565$	−11.6870	−0.491676
$566$	0	0
$567$	4.67440	0.196306
$568$	0	0
$569$	−24.3712	−1.02169	−0.510846	−	0.859672i	$-0.670668\pi$
$569$	−24.3712	−1.02169	−0.510846	+	0.859672i	$0.670668\pi$
$570$	0	0
$571$	10.2412	0.428580	0.214290	−	0.976770i	$-0.431256\pi$
$571$	10.2412	0.428580	0.214290	+	0.976770i	$0.431256\pi$
$572$	0	0
$573$	17.3951	0.726689
$574$	0	0
$575$	−1.29965	−0.0541992
$576$	0	0
$577$	33.8663	1.40987	0.704936	−	0.709271i	$-0.250976\pi$
$577$	33.8663	1.40987	0.704936	+	0.709271i	$0.250976\pi$
$578$	0	0
$579$	−18.8082	−0.781641
$580$	0	0
$581$	−56.9479	−2.36259
$582$	0	0
$583$	2.32707	0.0963775
$584$	0	0
$585$	−4.13882	−0.171119
$586$	0	0
$587$	−15.4940	−0.639504	−0.319752	−	0.947501i	$-0.603600\pi$
$587$	−15.4940	−0.639504	−0.319752	+	0.947501i	$0.603600\pi$
$588$	0	0
$589$	−3.02117	−0.124485
$590$	0	0
$591$	−9.50851	−0.391128
$592$	0	0
$593$	11.1997	0.459916	0.229958	−	0.973201i	$-0.426141\pi$
$593$	11.1997	0.459916	0.229958	+	0.973201i	$0.426141\pi$
$594$	0	0
$595$	34.4962	1.41421
$596$	0	0
$597$	21.1613	0.866076
$598$	0	0
$599$	−43.3751	−1.77226	−0.886129	−	0.463439i	$-0.846615\pi$
$599$	−43.3751	−1.77226	−0.886129	+	0.463439i	$0.846615\pi$
$600$	0	0
$601$	1.92124	0.0783689	0.0391844	−	0.999232i	$-0.487524\pi$
$601$	1.92124	0.0783689	0.0391844	+	0.999232i	$0.487524\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	0	0
$605$	−10.4239	−0.423792
$606$	0	0
$607$	−35.7420	−1.45072	−0.725361	−	0.688368i	$-0.758327\pi$
$607$	−35.7420	−1.45072	−0.725361	+	0.688368i	$0.758327\pi$
$608$	0	0
$609$	9.86162	0.399613
$610$	0	0
$611$	−3.85810	−0.156082
$612$	0	0
$613$	−14.4417	−0.583296	−0.291648	−	0.956526i	$-0.594204\pi$
$613$	−14.4417	−0.583296	−0.291648	+	0.956526i	$0.594204\pi$
$614$	0	0
$615$	−4.50124	−0.181507
$616$	0	0
$617$	17.5406	0.706156	0.353078	−	0.935594i	$-0.385135\pi$
$617$	17.5406	0.706156	0.353078	+	0.935594i	$0.385135\pi$
$618$	0	0
$619$	23.7014	0.952641	0.476320	−	0.879272i	$-0.341970\pi$
$619$	23.7014	0.952641	0.476320	+	0.879272i	$0.341970\pi$
$620$	0	0
$621$	1.29965	0.0521532
$622$	0	0
$623$	−56.0889	−2.24715
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	3.25074	0.129822
$628$	0	0
$629$	24.3508	0.970930
$630$	0	0
$631$	14.8143	0.589747	0.294873	−	0.955536i	$-0.404723\pi$
$631$	14.8143	0.589747	0.294873	+	0.955536i	$0.404723\pi$
$632$	0	0
$633$	5.73164	0.227812
$634$	0	0
$635$	10.4527	0.414803
$636$	0	0
$637$	−61.4617	−2.43520
$638$	0	0
$639$	6.64006	0.262677
$640$	0	0
$641$	−19.1460	−0.756223	−0.378112	−	0.925760i	$-0.623426\pi$
$641$	−19.1460	−0.756223	−0.378112	+	0.925760i	$0.623426\pi$
$642$	0	0
$643$	36.0238	1.42064	0.710320	−	0.703878i	$-0.248550\pi$
$643$	36.0238	1.42064	0.710320	+	0.703878i	$0.248550\pi$
$644$	0	0
$645$	−0.367478	−0.0144694
$646$	0	0
$647$	−10.1860	−0.400454	−0.200227	−	0.979750i	$-0.564168\pi$
$647$	−10.1860	−0.400454	−0.200227	+	0.979750i	$0.564168\pi$
$648$	0	0
$649$	−4.65942	−0.182898
$650$	0	0
$651$	3.29736	0.129234
$652$	0	0
$653$	12.7039	0.497142	0.248571	−	0.968614i	$-0.420039\pi$
$653$	12.7039	0.497142	0.248571	+	0.968614i	$0.420039\pi$
$654$	0	0
$655$	−9.37981	−0.366500
$656$	0	0
$657$	−6.39258	−0.249398
$658$	0	0
$659$	−29.5598	−1.15149	−0.575743	−	0.817630i	$-0.695287\pi$
$659$	−29.5598	−1.15149	−0.575743	+	0.817630i	$0.695287\pi$
$660$	0	0
$661$	−31.4289	−1.22244	−0.611220	−	0.791460i	$-0.709321\pi$
$661$	−31.4289	−1.22244	−0.611220	+	0.791460i	$0.709321\pi$
$662$	0	0
$663$	30.5437	1.18622
$664$	0	0
$665$	20.0199	0.776337
$666$	0	0
$667$	2.74188	0.106166
$668$	0	0
$669$	−12.6470	−0.488962
$670$	0	0
$671$	1.73272	0.0668910
$672$	0	0
$673$	16.0571	0.618956	0.309478	−	0.950907i	$-0.399846\pi$
$673$	16.0571	0.618956	0.309478	+	0.950907i	$0.399846\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−44.5968	−1.71400	−0.856998	−	0.515320i	$-0.827673\pi$
$677$	−44.5968	−1.71400	−0.856998	+	0.515320i	$0.827673\pi$
$678$	0	0
$679$	89.8969	3.44993
$680$	0	0
$681$	−10.3426	−0.396330
$682$	0	0
$683$	−6.69603	−0.256217	−0.128108	−	0.991760i	$-0.540891\pi$
$683$	−6.69603	−0.256217	−0.128108	+	0.991760i	$0.540891\pi$
$684$	0	0
$685$	−9.30026	−0.355345
$686$	0	0
$687$	6.47195	0.246920
$688$	0	0
$689$	12.6894	0.483426
$690$	0	0
$691$	39.5645	1.50511	0.752553	−	0.658532i	$-0.228822\pi$
$691$	39.5645	1.50511	0.752553	+	0.658532i	$0.228822\pi$
$692$	0	0
$693$	−3.54792	−0.134774
$694$	0	0
$695$	−19.9648	−0.757308
$696$	0	0
$697$	33.2183	1.25823
$698$	0	0
$699$	−25.3946	−0.960513
$700$	0	0
$701$	−28.7678	−1.08654	−0.543272	−	0.839557i	$-0.682815\pi$
$701$	−28.7678	−1.08654	−0.543272	+	0.839557i	$0.682815\pi$
$702$	0	0
$703$	14.1320	0.532998
$704$	0	0
$705$	−0.932174	−0.0351077
$706$	0	0
$707$	−9.73545	−0.366139
$708$	0	0
$709$	−8.32373	−0.312604	−0.156302	−	0.987709i	$-0.549957\pi$
$709$	−8.32373	−0.312604	−0.156302	+	0.987709i	$0.549957\pi$
$710$	0	0
$711$	5.62223	0.210850
$712$	0	0
$713$	0.916784	0.0343338
$714$	0	0
$715$	3.14140	0.117482
$716$	0	0
$717$	21.7439	0.812040
$718$	0	0
$719$	26.2136	0.977604	0.488802	−	0.872395i	$-0.337434\pi$
$719$	26.2136	0.977604	0.488802	+	0.872395i	$0.337434\pi$
$720$	0	0
$721$	−1.08365	−0.0403573
$722$	0	0
$723$	−28.2735	−1.05150
$724$	0	0
$725$	−2.10971	−0.0783525
$726$	0	0
$727$	34.0206	1.26176	0.630878	−	0.775882i	$-0.282695\pi$
$727$	34.0206	1.26176	0.630878	+	0.775882i	$0.282695\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.71192	0.100304
$732$	0	0
$733$	21.6684	0.800341	0.400170	−	0.916441i	$-0.368951\pi$
$733$	21.6684	0.800341	0.400170	+	0.916441i	$0.368951\pi$
$734$	0	0
$735$	−14.8500	−0.547752
$736$	0	0
$737$	−0.759009	−0.0279585
$738$	0	0
$739$	−0.686064	−0.0252373	−0.0126186	−	0.999920i	$-0.504017\pi$
$739$	−0.686064	−0.0252373	−0.0126186	+	0.999920i	$0.504017\pi$
$740$	0	0
$741$	17.7260	0.651182
$742$	0	0
$743$	−5.11580	−0.187681	−0.0938403	−	0.995587i	$-0.529914\pi$
$743$	−5.11580	−0.187681	−0.0938403	+	0.995587i	$0.529914\pi$
$744$	0	0
$745$	6.27275	0.229816
$746$	0	0
$747$	−12.1829	−0.445750
$748$	0	0
$749$	−17.4927	−0.639169
$750$	0	0
$751$	12.1818	0.444521	0.222261	−	0.974987i	$-0.428656\pi$
$751$	12.1818	0.444521	0.222261	+	0.974987i	$0.428656\pi$
$752$	0	0
$753$	−15.6216	−0.569285
$754$	0	0
$755$	6.68496	0.243291
$756$	0	0
$757$	−3.59744	−0.130751	−0.0653755	−	0.997861i	$-0.520825\pi$
$757$	−3.59744	−0.130751	−0.0653755	+	0.997861i	$0.520825\pi$
$758$	0	0
$759$	−0.986448	−0.0358058
$760$	0	0
$761$	−52.9557	−1.91964	−0.959822	−	0.280611i	$-0.909463\pi$
$761$	−52.9557	−1.91964	−0.959822	+	0.280611i	$0.909463\pi$
$762$	0	0
$763$	71.1262	2.57494
$764$	0	0
$765$	7.37981	0.266818
$766$	0	0
$767$	−25.4075	−0.917410
$768$	0	0
$769$	−39.3708	−1.41975	−0.709874	−	0.704329i	$-0.751248\pi$
$769$	−39.3708	−1.41975	−0.709874	+	0.704329i	$0.751248\pi$
$770$	0	0
$771$	−18.5901	−0.669505
$772$	0	0
$773$	−15.9266	−0.572841	−0.286420	−	0.958104i	$-0.592465\pi$
$773$	−15.9266	−0.572841	−0.286420	+	0.958104i	$0.592465\pi$
$774$	0	0
$775$	−0.705407	−0.0253390
$776$	0	0
$777$	−15.4239	−0.553330
$778$	0	0
$779$	19.2782	0.690714
$780$	0	0
$781$	−5.03987	−0.180341
$782$	0	0
$783$	2.10971	0.0753948
$784$	0	0
$785$	9.82160	0.350548
$786$	0	0
$787$	−17.2464	−0.614766	−0.307383	−	0.951586i	$-0.599453\pi$
$787$	−17.2464	−0.614766	−0.307383	+	0.951586i	$0.599453\pi$
$788$	0	0
$789$	2.37571	0.0845774
$790$	0	0
$791$	−54.6297	−1.94241
$792$	0	0
$793$	9.44839	0.335522
$794$	0	0
$795$	3.06593	0.108737
$796$	0	0
$797$	29.9130	1.05957	0.529786	−	0.848132i	$-0.322272\pi$
$797$	29.9130	1.05957	0.529786	+	0.848132i	$0.322272\pi$
$798$	0	0
$799$	6.87927	0.243371
$800$	0	0
$801$	−11.9992	−0.423969
$802$	0	0
$803$	4.85203	0.171224
$804$	0	0
$805$	−6.07510	−0.214119
$806$	0	0
$807$	1.47818	0.0520344
$808$	0	0
$809$	−50.6166	−1.77958	−0.889792	−	0.456367i	$-0.849151\pi$
$809$	−50.6166	−1.77958	−0.889792	+	0.456367i	$0.849151\pi$
$810$	0	0
$811$	21.1459	0.742532	0.371266	−	0.928527i	$-0.378924\pi$
$811$	21.1459	0.742532	0.371266	+	0.928527i	$0.378924\pi$
$812$	0	0
$813$	−2.44807	−0.0858576
$814$	0	0
$815$	11.0419	0.386780
$816$	0	0
$817$	1.57386	0.0550624
$818$	0	0
$819$	−19.3465	−0.676022
$820$	0	0
$821$	−28.0319	−0.978319	−0.489160	−	0.872194i	$-0.662696\pi$
$821$	−28.0319	−0.978319	−0.489160	+	0.872194i	$0.662696\pi$
$822$	0	0
$823$	−2.73850	−0.0954583	−0.0477291	−	0.998860i	$-0.515198\pi$
$823$	−2.73850	−0.0954583	−0.0477291	+	0.998860i	$0.515198\pi$
$824$	0	0
$825$	0.759009	0.0264253
$826$	0	0
$827$	−22.4440	−0.780455	−0.390228	−	0.920718i	$-0.627604\pi$
$827$	−22.4440	−0.780455	−0.390228	+	0.920718i	$0.627604\pi$
$828$	0	0
$829$	25.0350	0.869500	0.434750	−	0.900551i	$-0.356837\pi$
$829$	25.0350	0.869500	0.434750	+	0.900551i	$0.356837\pi$
$830$	0	0
$831$	−3.66859	−0.127262
$832$	0	0
$833$	109.591	3.79709
$834$	0	0
$835$	6.38964	0.221123
$836$	0	0
$837$	0.705407	0.0243824
$838$	0	0
$839$	−40.6630	−1.40384	−0.701922	−	0.712254i	$-0.747675\pi$
$839$	−40.6630	−1.40384	−0.701922	+	0.712254i	$0.747675\pi$
$840$	0	0
$841$	−24.5491	−0.846522
$842$	0	0
$843$	−22.6308	−0.779447
$844$	0	0
$845$	4.12983	0.142070
$846$	0	0
$847$	−48.7255	−1.67423
$848$	0	0
$849$	−2.84696	−0.0977073
$850$	0	0
$851$	−4.28840	−0.147004
$852$	0	0
$853$	37.8575	1.29622	0.648108	−	0.761548i	$-0.275561\pi$
$853$	37.8575	1.29622	0.648108	+	0.761548i	$0.275561\pi$
$854$	0	0
$855$	4.28287	0.146471
$856$	0	0
$857$	38.2037	1.30501	0.652507	−	0.757783i	$-0.273717\pi$
$857$	38.2037	1.30501	0.652507	+	0.757783i	$0.273717\pi$
$858$	0	0
$859$	−36.8680	−1.25792	−0.628960	−	0.777438i	$-0.716519\pi$
$859$	−36.8680	−1.25792	−0.628960	+	0.777438i	$0.716519\pi$
$860$	0	0
$861$	−21.0406	−0.717062
$862$	0	0
$863$	−20.0508	−0.682536	−0.341268	−	0.939966i	$-0.610856\pi$
$863$	−20.0508	−0.682536	−0.341268	+	0.939966i	$0.610856\pi$
$864$	0	0
$865$	−6.13693	−0.208662
$866$	0	0
$867$	−37.4616	−1.27226
$868$	0	0
$869$	−4.26733	−0.144759
$870$	0	0
$871$	−4.13882	−0.140239
$872$	0	0
$873$	19.2317	0.650896
$874$	0	0
$875$	4.67440	0.158024
$876$	0	0
$877$	37.6175	1.27025	0.635126	−	0.772409i	$-0.280948\pi$
$877$	37.6175	1.27025	0.635126	+	0.772409i	$0.280948\pi$
$878$	0	0
$879$	−2.75351	−0.0928737
$880$	0	0
$881$	−39.6012	−1.33420	−0.667100	−	0.744968i	$-0.732464\pi$
$881$	−39.6012	−1.33420	−0.667100	+	0.744968i	$0.732464\pi$
$882$	0	0
$883$	26.8409	0.903269	0.451635	−	0.892203i	$-0.350841\pi$
$883$	26.8409	0.903269	0.451635	+	0.892203i	$0.350841\pi$
$884$	0	0
$885$	−6.13882	−0.206354
$886$	0	0
$887$	11.2464	0.377616	0.188808	−	0.982014i	$-0.439538\pi$
$887$	11.2464	0.377616	0.188808	+	0.982014i	$0.439538\pi$
$888$	0	0
$889$	48.8601	1.63872
$890$	0	0
$891$	−0.759009	−0.0254278
$892$	0	0
$893$	3.99238	0.133600
$894$	0	0
$895$	−5.95000	−0.198887
$896$	0	0
$897$	−5.37903	−0.179600
$898$	0	0
$899$	1.48820	0.0496343
$900$	0	0
$901$	−22.6260	−0.753782
$902$	0	0
$903$	−1.71774	−0.0571628
$904$	0	0
$905$	15.2055	0.505449
$906$	0	0
$907$	−30.5933	−1.01583	−0.507916	−	0.861406i	$-0.669584\pi$
$907$	−30.5933	−1.01583	−0.507916	+	0.861406i	$0.669584\pi$
$908$	0	0
$909$	−2.08272	−0.0690793
$910$	0	0
$911$	12.9191	0.428028	0.214014	−	0.976831i	$-0.431346\pi$
$911$	12.9191	0.428028	0.214014	+	0.976831i	$0.431346\pi$
$912$	0	0
$913$	9.24695	0.306029
$914$	0	0
$915$	2.28287	0.0754694
$916$	0	0
$917$	−43.8450	−1.44789
$918$	0	0
$919$	−33.9746	−1.12072	−0.560359	−	0.828250i	$-0.689337\pi$
$919$	−33.9746	−1.12072	−0.560359	+	0.828250i	$0.689337\pi$
$920$	0	0
$921$	−2.39592	−0.0789484
$922$	0	0
$923$	−27.4820	−0.904581
$924$	0	0
$925$	3.29965	0.108492
$926$	0	0
$927$	−0.231827	−0.00761419
$928$	0	0
$929$	11.3738	0.373163	0.186582	−	0.982439i	$-0.440259\pi$
$929$	11.3738	0.373163	0.186582	+	0.982439i	$0.440259\pi$
$930$	0	0
$931$	63.6008	2.08443
$932$	0	0
$933$	−24.6362	−0.806554
$934$	0	0
$935$	−5.60135	−0.183184
$936$	0	0
$937$	−20.9717	−0.685116	−0.342558	−	0.939497i	$-0.611293\pi$
$937$	−20.9717	−0.685116	−0.342558	+	0.939497i	$0.611293\pi$
$938$	0	0
$939$	−12.3413	−0.402742
$940$	0	0
$941$	−24.2330	−0.789972	−0.394986	−	0.918687i	$-0.629251\pi$
$941$	−24.2330	−0.789972	−0.394986	+	0.918687i	$0.629251\pi$
$942$	0	0
$943$	−5.85004	−0.190504
$944$	0	0
$945$	−4.67440	−0.152058
$946$	0	0
$947$	−4.59741	−0.149396	−0.0746979	−	0.997206i	$-0.523799\pi$
$947$	−4.59741	−0.149396	−0.0746979	+	0.997206i	$0.523799\pi$
$948$	0	0
$949$	26.4577	0.858854
$950$	0	0
$951$	−12.7136	−0.412268
$952$	0	0
$953$	−1.95779	−0.0634191	−0.0317095	−	0.999497i	$-0.510095\pi$
$953$	−1.95779	−0.0634191	−0.0317095	+	0.999497i	$0.510095\pi$
$954$	0	0
$955$	−17.3951	−0.562891
$956$	0	0
$957$	−1.60129	−0.0517623
$958$	0	0
$959$	−43.4732	−1.40382
$960$	0	0
$961$	−30.5024	−0.983948
$962$	0	0
$963$	−3.74223	−0.120592
$964$	0	0
$965$	18.8082	0.605456
$966$	0	0
$967$	33.4688	1.07628	0.538141	−	0.842855i	$-0.319127\pi$
$967$	33.4688	1.07628	0.538141	+	0.842855i	$0.319127\pi$
$968$	0	0
$969$	−31.6068	−1.01536
$970$	0	0
$971$	32.3070	1.03678	0.518390	−	0.855144i	$-0.326532\pi$
$971$	32.3070	1.03678	0.518390	+	0.855144i	$0.326532\pi$
$972$	0	0
$973$	−93.3236	−2.99182
$974$	0	0
$975$	4.13882	0.132548
$976$	0	0
$977$	−42.4710	−1.35877	−0.679383	−	0.733784i	$-0.737753\pi$
$977$	−42.4710	−1.35877	−0.679383	+	0.733784i	$0.737753\pi$
$978$	0	0
$979$	9.10747	0.291076
$980$	0	0
$981$	15.2161	0.485812
$982$	0	0
$983$	49.1097	1.56635	0.783177	−	0.621798i	$-0.213598\pi$
$983$	49.1097	1.56635	0.783177	+	0.621798i	$0.213598\pi$
$984$	0	0
$985$	9.50851	0.302966
$986$	0	0
$987$	−4.35736	−0.138696
$988$	0	0
$989$	−0.477593	−0.0151866
$990$	0	0
$991$	−18.5728	−0.589985	−0.294992	−	0.955500i	$-0.595317\pi$
$991$	−18.5728	−0.589985	−0.294992	+	0.955500i	$0.595317\pi$
$992$	0	0
$993$	16.4136	0.520871
$994$	0	0
$995$	−21.1613	−0.670859
$996$	0	0
$997$	−3.87937	−0.122861	−0.0614304	−	0.998111i	$-0.519566\pi$
$997$	−3.87937	−0.122861	−0.0614304	+	0.998111i	$0.519566\pi$
$998$	0	0
$999$	−3.29965	−0.104396

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.f.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.f.1.6	✓	6	1.1	even	1	trivial

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Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +4.67440 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} +4.67440 q^{7} +1.00000 q^{9} -0.759009 q^{11} -4.13882 q^{13} -1.00000 q^{15} +7.37981 q^{17} +4.28287 q^{19} -4.67440 q^{21} -1.29965 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.10971 q^{29} -0.705407 q^{31} +0.759009 q^{33} +4.67440 q^{35} +3.29965 q^{37} +4.13882 q^{39} +4.50124 q^{41} +0.367478 q^{43} +1.00000 q^{45} +0.932174 q^{47} +14.8500 q^{49} -7.37981 q^{51} -3.06593 q^{53} -0.759009 q^{55} -4.28287 q^{57} +6.13882 q^{59} -2.28287 q^{61} +4.67440 q^{63} -4.13882 q^{65} +1.00000 q^{67} +1.29965 q^{69} +6.64006 q^{71} -6.39258 q^{73} -1.00000 q^{75} -3.54792 q^{77} +5.62223 q^{79} +1.00000 q^{81} -12.1829 q^{83} +7.37981 q^{85} +2.10971 q^{87} -11.9992 q^{89} -19.3465 q^{91} +0.705407 q^{93} +4.28287 q^{95} +19.2317 q^{97} -0.759009 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{5} + q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 6 * q^5 + q^7 + 6 * q^9	\( 6 q - 6 q^{3} + 6 q^{5} + q^{7} + 6 q^{9} - 7 q^{11} + 7 q^{13} - 6 q^{15} + 10 q^{17} - 3 q^{19} - q^{21} - 4 q^{23} + 6 q^{25} - 6 q^{27} + 9 q^{29} + 3 q^{31} + 7 q^{33} + q^{35} + 16 q^{37} - 7 q^{39} + 7 q^{41} + 3 q^{43} + 6 q^{45} + q^{47} + 15 q^{49} - 10 q^{51} + 7 q^{53} - 7 q^{55} + 3 q^{57} + 5 q^{59} + 15 q^{61} + q^{63} + 7 q^{65} + 6 q^{67} + 4 q^{69} - 12 q^{71} + 12 q^{73} - 6 q^{75} + 9 q^{77} + 9 q^{79} + 6 q^{81} - 2 q^{83} + 10 q^{85} - 9 q^{87} + 10 q^{89} - 5 q^{91} - 3 q^{93} - 3 q^{95} + 37 q^{97} - 7 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^5 + q^7 + 6 * q^9 - 7 * q^11 + 7 * q^13 - 6 * q^15 + 10 * q^17 - 3 * q^19 - q^21 - 4 * q^23 + 6 * q^25 - 6 * q^27 + 9 * q^29 + 3 * q^31 + 7 * q^33 + q^35 + 16 * q^37 - 7 * q^39 + 7 * q^41 + 3 * q^43 + 6 * q^45 + q^47 + 15 * q^49 - 10 * q^51 + 7 * q^53 - 7 * q^55 + 3 * q^57 + 5 * q^59 + 15 * q^61 + q^63 + 7 * q^65 + 6 * q^67 + 4 * q^69 - 12 * q^71 + 12 * q^73 - 6 * q^75 + 9 * q^77 + 9 * q^79 + 6 * q^81 - 2 * q^83 + 10 * q^85 - 9 * q^87 + 10 * q^89 - 5 * q^91 - 3 * q^93 - 3 * q^95 + 37 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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